The goal of the paper is to define Hochschild and cyclic homology for bornological coarse spaces, i.e., lax symmetric monoidal functors \({{\,\mathrm{\mathcal {X}HH}\,}}_{}^G\) and \({{\,\mathrm{\mathcal {X}HC}\,}}_{}^G\) from the category \(G\mathbf {BornCoarse}\) of equivariant bornological coarse spaces to the cocomplete stable \(\infty \)-category \(\mathbf {Ch}_\infty \) of chain complexes reminiscent of the classical Hochschild and cyclic homology. We investigate relations to coarse algebraic K-theory \(\mathcal {X}K^G_{}\) and to coarse ordinary homology \({{\,\mathrm{\mathcal {X}H}\,}}^G\) by constructing a trace-like natural transformation \(\mathcal {X}K_{}^G\rightarrow {{\,\mathrm{\mathcal {X}H}\,}}^G\) that factors through coarse Hochschild (and cyclic) homology. We further compare the forget-control map for \({{\,\mathrm{\mathcal {X}HH}\,}}_{}^G\) with the associated generalized assembly map.

本文的目的是为强生的粗糙空间定义Hochschild和循环同源性，即松弛对称的单向子函子\（{{\，\ mathrm {\ mathcal {X} HH} \，}} _ {} ^ G \）和\（{{\\\ mathrm {\ mathcal {X} HC} \，}} _ {} ^ G \）从等变出生学粗糙空间\（G \ mathbf {BornCoarse} \）到共完成稳定\（\ infty \）-类别\（\ mathbf {Ch} _ \ infty \）让人联想到经典的Hochschild和循环同源性。我们研究与粗代数K理论\（\ mathcal {X} K ^ G _ {} \）和与粗普通同源性 \（{{\，\ mathrm {\ mathcal {X} H} \，}} ^ G \）通过构造类似痕迹的自然变换\（\ mathcal {X} K _ {} ^ G \ rightarrow {{\，\ mathrm {\ mathcal {X} H} \，}} ^ G \）来通过粗糙的Hochschild（和循环）同源性。我们进一步将\（{{\，\ mathrm {\ mathcal {X} HH} \，}} _ {} ^ G \）的忘记控制图与关联的广义程序集图进行比较。